27/05/2019
IPCC (2012)
The marginal distribution of temperature extremes is changing -
Is the dependence also changing?
If so by how much?
What does this mean for events like heatwaves?
Marginals
Dependence
Observations
Climate Models
Let \(M\) hottest 3-day average temperature in a year: \[
\mathbb{P}(M < z) = \exp \left\{ - \left[ 1 + \xi \left(\dfrac{z-\mu}{\sigma}\right) \right]_+^{-1 / \xi} \right\},\] where \([v]_+ = \max \left\lbrace 0,v \right\rbrace\), \(\mu \in \mathbb{R}\), \(\sigma \in \mathbb{R}^+\) and \(\xi \in \mathbb{R}\).
(Fisher and Tippett 1928, Gnendenko 1943)
Address the trend in changing temperature by fitting temporal covariates to our parameters
Transform the GEV marginals to common GEV(1,1,1) marignal distributions:
\(\mathbb{P}(M < z) = \exp(-1/z)\)
Now dependence!
Assume the distribution of \((M_{i}, M_{j})\) can be approximated with a bivariate extreme value theory distribution, then
\[\mathbb{P}\left( M_{i} \leq z_i,\, M_{j} \leq z_j \right) = \exp\left\{ - V_{ij} \left( \dfrac{-1}{\log F_i(z_i)}, \, \dfrac{-1}{\log F_j(z_j)}\right) \right\}\] where the exponent measure \(V_{ij}(a,b)\) is given by \[V_{ij}(a, b) = 2 \int_0^1 \max \left(\dfrac{w}{a}, \dfrac{1-w}{b} \right) \text{d}H_{ij}(w)\] and \(H\) is any distribution function on \([0,1]\) with expectation equal to 0.5.
Lots of parameteric options to specify form of \(V_{ij}(a, b)\), but
Want to avoid assuming that a parametric form that is constant in time
Want a solution that will work for lots of different pairs
so …
Nonparametric
\[V_{ij}(z_i, z_j) = \left(\frac{1}{z_i} + \frac{1}{z_j}\right)A\left(\frac{z_i}{|z_i + z_j|} + \frac{z_j}{|z_i + z_j|}\right)\]
C1: \(A(t)\) is continuous and convex for \(t \in [0,1]\)
C2: Bounds on the function of \(\max \{(1 - t), t \} \leq A(t) \leq 1\)
C3: Boundary conditions of \(A(0) = A(1) = 1\)
Model background:
22 Global Climate Model and Regional Climate Model combinations from EURO-CORDEX
0.11 degree resolation (upscale)
rotated-pole grid
1950-2005 with historical forcing
2006-2100 with RCP 8.5
Consider Annual Maxima
Early: 1951 - 2000
Middle: 2001 - 2050
Late: 2051 - 2100
Assume stationarity within each window (big assumption)
Fit pairwise distributions relative to a central location
\[\mathbb{P}\left(M_{i} \leq z, M_{j} \leq z\right) = \left[\mathbb{P}(M_{i} \leq z)\mathbb{P}(M_{j} \leq z) \right]^{A\left(1/2\right)}.\]
Early: 1951 - 2000
Strong initial evidence to suggest:
Fit a time-varying Pickands dependence function
Examine assumptions on the marginal distriubtion
Examine whether models are accurately representing heat extremes
Think about how to translate insights from extreme value statistics into insights about changing heat extremes
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